A man made a will in which he left ⁴/⁹ of his money to his wife and ⅖ of the remaining to his eldest child. The rest was to be shared equally among his four younger children. If each of the younger children received # 108,000, what was his wife's share?

Answer 1

To solve the problem, let's denote the total amount of money the man had as \( M \).

1. Calculate the amount left to the wife: The wife receives \( \frac{4}{9} \) of \( M \). Therefore, the amount received by the wife is: \[ \text{Wife's share} = \frac{4}{9} M \]

2. Calculate the remaining amount after the wife's share: The remaining amount after the wife’s share is: \[ M - \frac{4}{9}M = \frac{5}{9}M \]

3. Calculate the amount given to the eldest child: The eldest child receives \( \frac{2}{5} \) of the remaining amount: \[ \text{Eldest child's share} = \frac{2}{5} \times \frac{5}{9}M = \frac{2}{9} M \]

4. Calculate the total amount distributed to the wife and the eldest child: The total amount distributed to the wife and the eldest child is: \[ \text{Total amount to wife and eldest} = \frac{4}{9}M + \frac{2}{9}M = \frac{6}{9}M = \frac{2}{3}M \]

5. Calculate the amount left for the younger children: The remaining amount for the younger children is: \[ M - \frac{2}{3}M = \frac{1}{3}M \]

6. Share the remaining amount among the four younger children: This amount \( \frac{1}{3}M \) is shared equally among the 4 younger children: \[ \text{Share per younger child} = \frac{1}{4} \times \frac{1}{3}M = \frac{1}{12}M \]

7. Given that each younger child received 108,000, set up the equation: We know: \[ \frac{1}{12}M = 108,000 \]

8. Solve for \( M \): Multiply both sides by 12: \[ M = 108,000 \times 12 = 1,296,000 \]

9. Calculate the wife’s share: Now, substitute \( M \) back into the formula for the wife’s share: \[ \text{Wife's share} = \frac{4}{9}M = \frac{4}{9} \times 1,296,000 \] \[ = \frac{4 \times 1,296,000}{9} = \frac{5,184,000}{9} = 576,000 \]

Thus, the wife’s share is \( \boxed{576,000} \).